How to solve this 4th degree polynomial equation with complex coefficients numerically in Mathematica?

Question

I have a polynomial equation:

$-(a-ib)e^{(4\pi i/3)}(\sqrt{2}i+x^3/\sqrt{3})x- (a+ib) e^{(2\pi i/3)}(\sqrt{2}ix^3+1/\sqrt{3})=0$

(which in code is):

-(a - I b)Exp[4 π I/3](Sqrt[2] I+x^3/Sqrt[3])x - (a+I b)Exp[2 π I/3](Sqrt[2] I x^3+1/Sqrt[3])==0;

with the conditions:

$a^{2}+b^{2} \leq 1$,

$1/2 \geq a \geq -1$, and

$\sqrt{3}/2 \geq b \geq -\sqrt{3}/2$.

I need numerical solutions for this equation. I tried to solve this in Mathematica, but ended up with messy solutions. Also I couldn't impose conditions on $a$ and $b$. Please help me in solving this.

Answer 1

Here is one method of finding a numerical solution under your constraints

NMinimize[{Abs[-(a - I b) E^(4 Pi I/3) (Sqrt[2] I + x^3/Sqrt[3]) x -
  (a + I b) E^(2 Pi I/3) (Sqrt[2] I x^3 + 1/Sqrt[3])],
  {a^2 + b^2 <= 1, 1/2 >= a >= -1, Sqrt[3]/2 >= b >= -Sqrt[3]/2}}, {x, a, b}]

which gives you {x -> -1., a -> 0.330227, b -> -0.190657}

There are other solutions, but NMinimize is only returning one that it finds. If you want others you can try adding more conditions and see if it finds any solution in the smaller domain.

You should carefully read the help page for NMinimize and click on the "Details and Options" to see additional information. You want to be careful that the first part of what NMinimize returns, the minimum value found, is very close to zero. If that is not the case then it may have found a local, but non-zero minimum.

Answer 2

The solution of your equation can be obtained setting real and imaginary part equal to zero:

gln = ComplexExpand[{Re[#],Im[#]} &[-(a - I b) E^(4 Pi I/3) (Sqrt[2] I +x^3/Sqrt[3]) x - (a + I b) E^(2 Pi I/3) (Sqrt[2] I x^3 +1/Sqrt[3])], TargetFunctions -> {Re, Im}] 

Reduce (assuming all variables to be real) gives

sol = Reduce[gln == 0, x]

several conditions.

Now you have only to test which of these conditions are elements of the parameterspace {a,b}

Answer 3

Include the real condition in Solve. Here is your equation:

eqn = -(a - I b)Exp[4 π I/3](Sqrt[2] I+x^3/Sqrt[3])x - 
    (a+I b)Exp[2 π I/3](Sqrt[2] I x^3+1/Sqrt[3])==0;
eqn //TeXForm

$e^{-\frac{2 i \pi }{3}} x \left(\frac{x^3}{\sqrt{3}}+i \sqrt{2}\right) (-a+i b)-e^{\frac{2 i \pi }{3}} \left(\frac{1}{\sqrt{3}}+i \sqrt{2} x^3\right) (a+i b)=0$

and conditions:

conditions = a^2+b^2 <= 1 && -1 <= a <= 1/2 && -Sqrt[3]/2 <= b <= Sqrt[3]/2;

Use Solve, including reality conditions:

soln = Solve[eqn && (x | a | b) ∈ Reals];

Normal[soln] //ComplexExpand //Simplify //Column //TeXForm

$\begin{array}{l} \{a\to 0,b\to 0\} \\ \left\{a\to -\sqrt{3} b,x\to 1\right\} \\ \left\{a\to -\sqrt{3} b,x\to -1\right\} \\ \left\{a\to \frac{\sqrt{3} \left(31+22 \sqrt{2}\right) b}{9+6 \sqrt{2}},x\to -1-\sqrt{2}\right\} \\ \left\{a\to \frac{\sqrt{3} \left(22 \sqrt{2}-31\right) b}{6 \sqrt{2}-9},x\to 1-\sqrt{2}\right\} \\ \left\{a\to \frac{\left(5+4 \sqrt{2}\right) b}{\sqrt{3}},x\to \sqrt{2}-1\right\} \\ \left\{a\to \frac{\left(5-4 \sqrt{2}\right) b}{\sqrt{3}},x\to 1+\sqrt{2}\right\} \\ \end{array}$

Imposing your other conditions will just convert these infinite line regions into finite line regions. For example, the second solution corresponds to:

ComplexExpand @ Normal[soln[[2]]]

{a -> -Sqrt[3] b, x -> 1}

and:

RegionEqual[
    Line[{{-Sqrt[3]/2, 1/2}, {1/2, -1/(2Sqrt[3])}}],
    ImplicitRegion[a == -Sqrt[3] b && conditions, {a,b}]
]

True

Addendum

If there is no reality condition on x, then just do:

soln = Solve[eqn, x];

and plug in any values of a and b that satisfy your conditions. For example:

soln /. {a->.3, b->.1}

{{x -> -2.6158 - 0.635753 I}, {x -> -0.665718 + 0.958755 I}, {x -> 0.36097 + 0.0877316 I}, {x -> 0.488639 - 0.70373 I}}